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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 405600.df
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
405600.df1 | 405600df4 | \([0, -1, 0, -5308008, -4704550488]\) | \(428320044872/73125\) | \(2823683265000000000\) | \([2]\) | \(12386304\) | \(2.5469\) | |
405600.df2 | 405600df2 | \([0, -1, 0, -2266008, 1269177012]\) | \(33324076232/1285245\) | \(49629057065640000000\) | \([2]\) | \(12386304\) | \(2.5469\) | \(\Gamma_0(N)\)-optimal* |
405600.df3 | 405600df1 | \([0, -1, 0, -364758, -57895488]\) | \(1111934656/342225\) | \(1651854710025000000\) | \([2, 2]\) | \(6193152\) | \(2.2004\) | \(\Gamma_0(N)\)-optimal* |
405600.df4 | 405600df3 | \([0, -1, 0, 1008367, -391564863]\) | \(367061696/426465\) | \(-131741766411840000000\) | \([2]\) | \(12386304\) | \(2.5469\) |
Rank
sage: E.rank()
The elliptic curves in class 405600.df have rank \(1\).
Complex multiplication
The elliptic curves in class 405600.df do not have complex multiplication.Modular form 405600.2.a.df
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.